I need to get rid of water excess in my vacuum chamber, and for that there is the procedure of baking. In order to do that there are several things that one needs to consider, the power, heat load, type of heat tape to use etc. Since I've never done this before, I hope that some of you could share their advice on that subject.
[Physics] How to properly bake a ultra high vacuum chamber
experimental-physicsheatvacuum
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The "lighter than air" material you referred to is a network of carbon tubes so the air can obviously get through this "material" – most of the volume occupied by this material is the "air" – much like it can get through clothes (e.g. wool), so you can't use this "material" as the envelope for a balloon.
More generally, there obviously can't be a solid that would be lighter than the air at normal conditions. The reason is that the distance between the atoms or molecules of a solid has to be much shorter – comparable to the Bohr radius – than the average distance between an air molecule and its closest neighbor. Even if you place the lightest possible atoms at these distances, the density will already exceed the density of air by orders of magnitude. There simply can't be light-as-air impenetrable solids made out of any atoms and it's easy to see why.
Your "many layers, gradually changing pressure" construction doesn't solve anything. You may equally well send the room in between the shells to zero without changing anything substantial so having many layers only changes one thing, the total thickness of the balloon's envelope. And to beat one atmosphere which is really a big pressure, one really needs a thick material. Just to be sure, 1 atmosphere is 100,000 pascals or so. It's 100,000 newtons per squared meter – or, equivalently, 1 squared meter of the material has to keep its flat shape if you place 10 tons on it (in normal gravity).
If you want a balloon with those 3,000 cubic meters (of the balloon my dad just flew with, a gift we gave him to the birthday), note that the vacuum only saves 4,000 kilograms or so. A 9-meters-radius balloon has the surface of 1,000 squared meters or so. For the balloon to be lighter than the air, the envelope must have a surface density of less than 4 kilograms per squared meters. Even if you used a hypothetical material as light as water, 1,000 kg per cubic meter, it would have to be just 4 millimeters thin. And it's hard to imagine that such a thin material may support 10 tons per squared meter.
The usual solution – put a lighter gas inside – is clearly simpler and better. The hydrogen is approximately 15 times lighter than the air. Helium is 7 times lighter than the air. Hot air at doubled absolute temperature is 2 times lighter than the normal air and so on. With some lighter air inside, one may naturally balance the pressures.
The answer is no, or at least it is in the classical vacuum sense. I also don't see a rationale for why creating a vacuum would require infinite energy.
An explicit construction is to use a solid-phase reactive chemical "getter" to eliminate (nearly) all gas molecules present; in experimental practice, virtually all man-made materials still outgas slightly, which means that for practical purposes a true vacuum is difficult to achieve. As an example, one of the highest vacuums made on Earth was at CERN, with a density of 1 molecule per $\text{cm}^3$. However, this inability to create perfect vacuum is a problem of material science, rather than a side effect of theoretical impossibility.
In interstellar space, vacuums can approach 1 molecule per liter, which for all practical intents and purposes is perfect vacuum.
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The usual procedure involves
If heating is not going to work alternatively use a sorption pump with either liquid nitrogen or liquid helium.